Transitive A 6-invariant k-arcs in PG(2, q)

Abstract

For q = pr with a prime p ≥ 7 such that \({q \equiv 1}\) or 19 (mod 30), the desarguesian projective plane PG(2, q) of order q has a unique conjugacy class of projectivity groups isomorphic to the alternating group A6 of degree 6. For a projectivity group \({\Gamma \cong A_6}\) of PG(2, q), we investigate the geometric properties of the (unique) Γ-orbit \({\mathcal{O}}\) of size 90 such that the 1-point stabilizer of Γ in its action on \({\mathcal O}\) is a cyclic group of order 4. Here \({\mathcal O}\) lies either in PG(2, q) or in PG(2, q2) according as 3 is a square or a non-square element in GF(q). We show that if q ≥ 349 and q ≠ 421, then \({\mathcal O}\) is a 90-arc, which turns out to be complete for q = 349, 409, 529, 601,661. Interestingly, \({\mathcal O}\) is the smallest known complete arc in PG(2,601) and in PG(2,661). Computations are carried out by MAGMA.